To factor the polynomial [tex]\( x^2 + 11x + 24 \)[/tex], you need to find two numbers that add to 11 and multiply to 24, not the other way around.
Here are the steps to factor the quadratic polynomial [tex]\( x^2 + 11x + 24 \)[/tex]:
1. Identify the coefficients: In the polynomial [tex]\( x^2 + 11x + 24 \)[/tex], the coefficients are:
- The leading coefficient [tex]\( a = 1 \)[/tex]
- The middle term coefficient [tex]\( b = 11 \)[/tex]
- The constant term [tex]\( c = 24 \)[/tex]
2. Set up the conditions for factoring:
- We need to find two numbers [tex]\( m \)[/tex] and [tex]\( n \)[/tex] such that:
- [tex]\( m + n = 11 \)[/tex] (they add to the middle term coefficient)
- [tex]\( m \cdot n = 24 \)[/tex] (they multiply to the constant term)
3. Find the numbers:
- We look for pairs of numbers that multiply to 24 and check if they add to 11:
- [tex]\( 1 \cdot 24 = 24 \)[/tex], but [tex]\( 1 + 24 = 25 \)[/tex]
- [tex]\( 2 \cdot 12 = 24 \)[/tex], but [tex]\( 2 + 12 = 14 \)[/tex]
- [tex]\( 3 \cdot 8 = 24 \)[/tex], and [tex]\( 3 + 8 = 11 \)[/tex] (this pair works)
- [tex]\( 4 \cdot 6 = 24 \)[/tex], but [tex]\( 4 + 6 = 10 \)[/tex]
Thus, the numbers 3 and 8 satisfy both conditions.
4. Rewrite the middle term using the found numbers:
- Rewrite [tex]\( 11x \)[/tex] as [tex]\( 3x + 8x \)[/tex]:
[tex]\[
x^2 + 11x + 24 = x^2 + 3x + 8x + 24
\][/tex]
5. Factor by grouping:
- Group the terms to factor by grouping:
[tex]\[
x^2 + 3x + 8x + 24 = x(x + 3) + 8(x + 3)
\][/tex]
- Factor out the common binomial [tex]\( x + 3 \)[/tex]:
[tex]\[
(x + 3)(x + 8)
\][/tex]
Therefore, the factored form of [tex]\( x^2 + 11x + 24 \)[/tex] is [tex]\( (x + 3)(x + 8) \)[/tex].
Conclusion:
The given statement "To factor the polynomial [tex]\( x^2 + 11x + 24 \)[/tex], find two numbers that add to 24, and multiply to 11" is False. The correct condition is to find two numbers that add to 11 and multiply to 24.
